Bidirectional rotating flow of nanofluid over a variable thickened stretching sheet with non-Fourier’s heat flux and non-Fick’s mass flux theory

The flow of nanofluid over a variable thickened stretching sheet is studied in this article. Non-Fourier’s heat flux and non-Fick’s mass flux are incorporated for heat and mass flow analysis. Silver (Ag) and Copper (Cu) are considered nanoparticles with water as base fluid. The resulting equations are transformed into the dimensionless form using similarity transformation and solved by RK-4 with the shooting method. The impact of the governing parameters on the dimensionless velocity, temperature, concentration, skin friction coefficient, streamlines, and finally isotherms are incorporated. It is observed that increment in power-law index parameter uplifts the fluid flow, heat, and mass transfer. The increase in the magnitude of skin friction coefficient in (x-direction) with wall thickness parameter is high for nanofluid containing silver nanoparticles as compared to copper nanoparticles.


Introduction
The problem of heat transfer enhancement in engineering and industrial applications is of high priority nowadays. Many applications that require fluid as a cooling agent, such as water, ethylene glycol, and low thermal conductivity oil, limit the heat transfer enhancement. To improve thermal efficiency, new types of fluids such as nanofluids are introduced. Masuda et al. [1] introduced ultrafine particles of Al 2 O 3 , SiO 2 and TiO 2 and observed change in thermal diffusivity and fluid viscosity. Nanofluids are described as a mixture composed of dispersed particles of nanometer size in the base fluid; the term initially introduced by Choi [ 1% by volume) magnifies the thermal conductivity of the fluid up to approximately two times [3]. Experimental studies of Xuan and Li [4] have shown that the small concentrations of nanoparticles can enhance the thermal conductivity of the suspensions by more than 20%. Akbarinia et al. [5] claimed that nanofluids are more stable and have reasonable viscosity and better properties for wetting, spreading, and dispersing on solid surfaces. Most used base fluids are water, ethylene glycol, and oil, while the common nanoparticles are gold, copper, silver, titanium, and silicon, or their oxides. Nanofluids are extensively used as coolants, lubricants, as well as in practical applications such as cooling and air conditioning, microelectronics, mobile computer processors, and so on. Recently, various researchers studied the behavior and characteristics of nanofluids both experimentally and mathematically. Experimental studies have shown that the addition of nanoparticles in conventional fluids improve the absorption efficiency of solar collectors' incident radiations (see refs. [6][7][8]). Also, the important application of nanofluid in microchannel has been discussed in detail by Chamkha et al. [9]. Nanofluids played a very important role in improving heat transfer at nuclear power plants.
Because of the high thermal conductivity of the nuclear reactor's nanofluid heat transfer capacity, it can be a thousand times larger than conventional fluids such as water [10]. Ferrofluids are formed in a liquid carrier by dispersing 10 nm ferromagnetic particles. Recent studies have shown that magnetic nanofluids can be used in many areas of magnetic separation of cells, contrast agent enhancement in magnetic resonance imaging (MRI), magnetic drug targeting, and magnetic hyperthermia (see refs. [11,12]). Additionally, various researchers have presented mathematical models of nanofluid, among them two are most commonly used. One proposed by Buongiorno [13] and the second by Tiwari and Das [14]. For the comparison between theoretical and experimental data, nanofluid thermal conductivity efficiency is being examined among the models. Various researchers such as [15][16][17][18][19][20][21] used Buongiorno's proposed mathematical nanofluid model [13] which involves the effects of Brownian motion and thermophoresis parameters. The nanofluid model proposed by Tiwari and Das [14] was also employed by several authors such as [22][23][24].
Recently, the implementation of Cattaneo-Christov model to study the mechanism of heat and mass flux is trending as various researchers adopt this model in problem formulation. The study of heat and mass flux cannot be ignored as the temperature and concentration differences occur within a system or within the systems. In all above-mentioned research, heat and mass transport phenomena are discussed by incorporating classical law of thermodynamics and heat conduction by Fourier [25]. The major drawback of using this law is the restriction of energy and concentration equations to be of parabolic type equations. The physical interpretation of this restriction is that any initial disturbance is rapidly experienced by the medium. Cattaneo [26] in 1948 modified the Fourier law by enforcing a thermal relaxation characteristic time term, called "thermal inertia" which is called Maxwell-Catteneo law. This modified law enables scientists to study heat/mass disturbance through the propagation of finite length wave. But the major difficulty in using this model is the existence of different thermal relaxation times for different materials. Keeping this fact in mind, a very useful time derivative model was developed for the effective heat transfer mechanism by Christov [27]. He replaced the time derivative with Oldroyd upper-convective derivative. The theory, thus, termed as Cattaneo-Christov heat and mass flux theory (CC-model). Haddad [28] investigated the thermal instability while analyzing the flow through Brinkman porous media using CC-model. Imtiaz et al. [29] performed the analysis of two-dimensional flow of non-Newtonian fluid over linearly stretching surface by incorporating the effects thermal relaxation time presented through CC-heat flux model. Mahmood et al. [30] utilized CC-model in heat flux analysis of Casson fluid flow over stretching surface with entropy generation effects. Negative impact of magnetic field on boundary layer flow nanofluid is noticed from the results. Kumar et al. [31] disclosed the influence of CC-model on Reiner-Philippoff flow on stretching surface in the presence of Ohmic heating and transverse magnetic field. Shah et al. [32] presented the effect of CC-heat flux model on Casson ferrofluid flow over the stretching surface in the presence of externally applied magnetic and electric fields. Shah et al. [33] also discussed the impact of CC-model on Darcy-Forchheimer flow of micropolar ferrofluid through porous media. Their study discloses that by increasing electric field strength velocity enhances whereas opposite behavior is noticed for microrotation parameter. Sajid et al. [34] reported the impact of CC-heat flux model on carbon nanotubes for Maxwell velocity slip and Smoluchowski temperature.
The analysis involving Newtonian and non-Newtonian fluid flow over stretched surface finds its application in various engineering processes which involves wire drawing, heat-treated materials travelling between a wind-up roll and a feed role or materials manufactured by extrusion, paper and glass fiber production, cooling of metallic sheets or electronic chips, drawing of plastic sheets, crystal growing, and many others. To our knowledge, nanofluid flow in the regimes of three-dimensional flow and rotating frame over variable thickened stretching sheet has not been explored yet. Purpose of present research is three-fold. Firstly, to formulate the bidirectional flow of silver and copper-water nanofluid bounded by stretching surface with variable thickness in rotating frame of reference. Secondly, to analyze heat and mass transfer with thermal and solutal relaxation effects. And thirdly, to solve the governing non-linear system by applying RK-4 with shooting method. Results are discussed for velocity, temperature, concentration and skin friction coefficients by plotting graphs.

Mathematical formulation
Here bidirectional flow of an incompressible nanofluid due to a stretching sheet of variable . Copper and silver nanoparticles with water are used as base fluid. We choose the Cartesian coordinate system such that the surface is aligned with the xy-plane and fluid is considered in the space z � 0. The surface is assumed to stretch in the x and y direction with stretching rate c. Also, the fluid rotates with constant angular velocity O about the z-axis (see Fig 1). T w is constant wall temperature and T 1 is the ambient temperature. Non-Fourier heat flux and non-Fick's mass flux are taken. The governing equations are The corresponding boundary conditions are where u, v and w depict the velocity components parallel to x−, y− and z−directions, v the kinematic viscosity of fluid, ρ fluid density, λ E heat relaxation time and λ C mass relaxation time. Type of the motion, shape of the surface and behavior of the boundary layer are controlled by the shape parameter n. Wall thickness parameter n is divided into three cases: the wall thickness parameter decreases for n>1, the surface is flat when n = 1 and the wall thickness parameter increases for n<1. As type of motion is controlled by this parameter so accelerated motion is represented by n>1, the linear motion by n =0 and the decelerated motion is expressed as n<1.
We consider the following transformations L ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðn þ 1Þ 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Here μ nf , ρ nf , (ρC p ) nf and k nf are the effective dynamic viscosity, density, heat capacity and thermal conductivity of the nanofluid [14] which are defined as Here φ represents the solid volume fraction of the nanoparticles and the subscripts s, f and nf are used for nano-solid-particles, base fluid, and thermo-physical properties of the nanofluid.
The conservation of mass is fulfilled and Eqs (2-6) are reduced to: f 0 ðLÞ ! 0; g 0 ðLÞ ! 0; yðLÞ ! 0; φðLÞ ! 0 as L ! 1: Here b ¼ A ffi ffi ffi ffi ffi ffi ffi ffi ffi ðnþ1Þc 2n f q is the wall thickness parameter. Writing Eqs (12-15) become: 1 Pr with boundary conditions Here Pr ¼ stands for Prandtl number, l ¼ Oðx þ yÞ nÀ 1 =c denotes rotation parameter, d ¼ l E cðx þ yÞ nÀ 1 denotes the thermal relaxation time and g ¼ l C cðx þ yÞ nÀ 1 denotes the mass relaxation time. Skin friction coefficients along the x− and y− directions are defined as follows: where the surface shear stresses τ wx and τ wy along x− and y− directions are Dimensionless skin friction coefficients are where ðRe x Þ 1=2 ¼ x ffi ffi ffi ffi ffi ffi ffi ffi c=n f q and ðRe y Þ 1=2 ¼ y ffi ffi ffi ffi ffi ffi ffi ffi c=n f q denote the local Reynolds number.

Results and discussion
In this study, the nonlinear and coupled Eqs (21) Here, the fluid velocity in both directions peters out with the augmented values of λ and ϕ. Physically, the higher value of rotation parameter corresponds to lower stretching rates (along x-direction) compared to the rotation rate. Moreover, it is also observed that the momentum boundary layer thickness reduced for Cu-water is slightly quicker than Ag-water case. Figs 6(A), 6(B) and 7(A), 7(B) illustrate the effects of β and ϕ on velocity profile. The growth in the fluid velocity is observed for solid volume fraction parameter while the larger values of wall thickness parameter correspond to small deformation due to stretching of wall and hence the velocity peters out in x-and y-directions. Fig 8(A) and 8(B) shows the impact of nanoparticles volume fraction ϕ and velocity power index n on temperature profile θ(ξ). An enhancement is observed for temperature profile θ(ξ) when nanoparticles volume fraction ϕ is increased. Physically, the thermal conductivity upsurges for higher value of the solid volume fraction parameter and consequently, fluid temperature increase. For nanoparticles volume fraction, the temperature of nanofluid containing copper nanoparticles was lower than that of containing silver nanoparticles. It is because the thermal conductivity of silver nanoparticles is high so the thickness of related boundary layer of Ag-water is greater than that of Cu-water nanofluid. It is further noticed that the thermal boundary layer is more thickens for an increasing value of velocity power index n. Fig 9(A) and 9(B) depicts the impact of λ and ϕ on the non-dimensional temperature field θ(ξ). It has been noticed that the temperature and thermal boundary layer escalate with increasing values of λ and ϕ. Because when rotation parameter is increased, the fluid resistivity increases which as a result enhances the fluid temperature. Fig 10(A) and 10(B) illustrate the influence of β and ϕ on temperature field θ(ξ). The growth in the fluid temperature is noticeably seen for solid volume fraction parameter while the larger values of wall thickness parameter upsurge the fluid temperature within boundary layer so that the related thermal layer become more thickens. Impact of thermal relaxation time parameter δ and ϕ on the temperature filed θ(ξ) is sketched in Fig 11(A) and 11(B). For Ag-and Cu-water, the temperature and related thermal layer thickness escalating with larger values of δ. As thermal relaxation parameter enlarges, the penetration depth of temperature decreases which enhances the temperature. The effects of n and ϕ on concentration F(ξ) is discussed in Fig 12(A) and 12(B). Large values of n and ϕ give rise into the nanoparticles concentration F(ξ) and the related boundary layer thickness.    values of skin friction coefficients (Re x ) 1/2 C fx and (Re y ) 1/2 C fy are negative since the fluid applies stress on the stretching wall (that causes the flow). Clearly, the magnitudes of (Re x ) 1/2 C fx and (Re y ) 1/2 C fy are an increasing function of ϕ. This fact can also be verified from the tabular values of skin friction factors (Re x ) 1/2 C fx and (Re y ) 1/2 C fy as presented in Tables 2 and 3 while Thermophysical properties of nanoparticles and water are given in Table 1.
The streamlines of the current analysis are drawn in Fig 19 for two different values of solid volume fractions (ϕ = 0,0.1). It is exposed that with the presence of solid volume fraction the rotating effects is more strengthen. The isotherms behavior outlines along the surface for two values of solid volume fractions have been explored in Fig 20. From the first plot of the isotherms, it is evidently seen that the isotherms are closed to small ξ and very closed too in the second plot in the case of ϕ = 0.1. This leads us to the result that for ϕ = 0.1, the accumulation of isotherms around the dimensionless variable ξ is increased.   In Table 4 we compared the results of f@(0) and g@(0) with existing literature in limiting sense. It is observed that obtained results are in good agreement with present findings.

Conclusions
The applications of stretching sheet are not only limited to industry but also finds its place in the geological studies where the fluid flows over submerged tectonic plate in rotating condition. Motivated by such applications the present article addresses three-dimensional rotational flow of nanofluid over nonlinearly stretching sheet of variable thickness. Tiwari and Das model is taken for nanofluid. The energy and concentration equations are modeled by incorporating CC-heat and mass flux effects. The mathematically modeled non-linear system of equations is solved with RK-4 cum shooting technique. The numerical and graphical outcomes for flow, energy and mass transport are obtained for various values of pertinent parameters. The computational investigation leads to some important results outlined as below: ■ Hydrodynamic boundary layer thickness is reduced due to larger nanoparticles volume fraction and rotation parameter ■ Thermal field θ(ξ) and the related boundary layer get increased due to the inclusion of rotation, nanoparticles volume fraction and thermal relaxation time.
■ Augmented β and Sc yield deescalated concentration F(ξ) and the related boundary layer.
■ Rise in β improves viscous drag force, from stretching surface, however, declines the same for solid volume fraction parameter ϕ.
■ For ϕ = 0.1, the accumulation of isotherms around the dimensionless variable ξ is increased.